Plane Couette Flow (Linear Profile)

Overview

Plane Couette flow is the shear flow set up between two parallel walls when one slides steadily past the other and there is no streamwise pressure gradient. It has an exact solution: a velocity that varies linearly across the gap, from zero on the stationary wall to the wall speed on the moving one. The profile is independent of the Reynolds number and of streamwise position, and the shear stress is uniform throughout, so the case exercises the solver's viscous (diffusion) operator, no-slip wall treatment, and tangential wall-motion mechanism. The computed profile is compared against the analytic linear distribution given by White (1994).

Problem Setup

The domain is the gap between two plates separated by a distance $h=1$ m. The lower wall is a stationary no-slip surface; the upper wall is a no-slip surface translating in its own plane at $U=34.03$ m/s, imposed through the solver's moving-frame mechanism so that the wall carries the fluid with it without any normal motion. The streamwise boundaries are a translational periodic pair, which makes the channel effectively infinite and removes any entrance effect, and the spanwise faces are symmetry planes so the simulation is quasi-two-dimensional. The fluid has constant density and constant viscosity, giving a Reynolds number $Re=\rho Uh/\mu \approx 1000$. The flow is laminar.

Quantities of Interest

The verification quantity is the streamwise velocity profile across the gap. With no pressure gradient the momentum balance reduces to $d^{2}u/d{y}^2=0$, whose solution under the no-slip conditions is the linear profile $u(y)=Uy/h$. The associated wall shear stress is the constant $\tau =\mu U/h$. The computed profile and its slope are compared against this analytic distribution to assess the diffusive momentum transport and the moving-wall boundary condition.